# An Efficient Structural Descriptor Sequence to Identify Graph Isomorphism and Graph Automorphism

## TL;DR

This paper introduces a new structural descriptor sequence based on neighborhood matrices to efficiently identify graph isomorphism and automorphism, demonstrating effectiveness on strongly regular datasets.

## Contribution

The paper proposes a novel graph invariant sequence derived from neighborhood matrices for improved graph isomorphism and automorphism detection.

## Key findings

- The descriptor sequence encodes complete structural information of graphs.
- The sequences are proven to be graph invariants.
- Effective on strongly regular datasets.

## Abstract

In this paper, we study the graph isomorphism and graph automorphism problems. We propose a novel technique to analyze graph isomorphism and graph automorphism. Further we handled some strongly regular datasets for prove the efficiency of our technique. The neighbourhood matrix $ \mathcal{NM}(G) $ was proposed in \cite {ALPaper} as a novel representation of graphs and was defined using the neighbourhood sets of the vertices. It was also shown that the matrix exhibits a bijection between the product of two well known graph matrices, namely the adjacency matrix and the Laplacian matrix. Further, in a recent work\cite{NM_SPath}, we introduced the sequence of matrices representing the powers of $\mathcal{NM}(G)$ and denoted it as $ \mathcal{NM}^{\{l\}}, 1\leq l \leq k(G)$ where $ k(G) $ is called the \textbf{iteration number}, $k(G)=\ceil*{\log_{2}diameter(G)} $. In this article we introduce a structural descriptor given by a sequence and clique sequence for any undirected unweighted simple graphs with help of the sequences of matrices $ NM^{\{l\}} $. The $ i^{th} $ element of structural descriptor sequence encodes the complete structural information of the graph from the vertex $ i\in V(G) $. The $ i^{th} $ element of clique sequence encodes the Maximal cliques on $ i $ vertices. The above sequences is shown to be a graph invariants and is used to study the graph isomorphism and automorphism problem.